Machine selection and part assignment analysis under annual budget constraint: a capital...

INTRODUCTION

The concept of outsourcing has been practiced for years by manufacturing industry. Outsourcing has both advantages and disadvantages. On the advantage side, organizations that practice outsourcing can increase their resource flexibility. This ability becomes an essential

need at present since the product life cycle tends to be shorter than that in the past. Outsourcing enables organizations to direct their limited workforce and capital toward crucial manufacturing processes, rather than having to ration them to sustain other not-too-crucial processes. Outsourcing also allows them to quickly adjust their product design and features to meet the dynamic market's demand. This is especially necessary when a higher degree of customization is being experienced. On the other side, outsourcing may cause organizations to lose control over their manufacturing efficiency. Late delivery, poor quality of parts, unreliable suppliers, etc. affect production planning and control. Organizations are forced to build up raw materials and parts inventory to smooth the manufacturing processes, resulting in high inventory cost. A reverse option to outsourcing is backward vertical integration in which organizations move toward the sources of raw materials and parts [6]. In this paper we place our emphasis on parts.

Choosing between backward vertical integration and outsourcing involves a "make-or-buy" decision-making analysis. Decision makers must study all benefits and costs of making the parts and buying them from suppliers. In several cases, an in-house manufacture of the parts requires additional machine investment. As the number of investment alternatives increases (when several machines are to be considered) and the amount of available capital is limited, capital budgeting decision making becomes more complex. The literature contains extensive discussions of mathematical approaches such as linear programming [1, 10, 11, 14], goal programming [5], nonlinear programming [11], and mixed-integer programming [2, 7] to capital budgeting problems. Also, analysis and evaluation techniques used in capital budgeting have been studied. Most recently, Lohmann and Baksh [8] compare the performance of several decision procedures based on net present value (NPV), internal rate of return (IRR), and payback period, both with risk-adjusted discount rate and risk-free discount rate, on capital budgeting decision making. Sarper [12] investigates the capital budgeting problem when cash flows and budgets are assumed to be uniformly distributed random variables (RVs). Using a chance-constrained programming technique to approximate uniformly distributed RVs into normally distributed ones, he shows that accurate results are obtained when compared with those obtained from a Monte-Carlo simulation. Evans and Forbes [4] discuss the preference of practitioners in using the IRR as an investment decision-making tool while business scholars prescribe the NPV as theoretically optimal. Chen [3] surveys project evaluation techniques used in capital budgeting. Statistical analyses show that discounted cash flow techniques are preferred to those of payback period and accounting rate of return, irrespective of investment types. Lohmann and Oakford [9] conduct a comparative investigation of an analytic model and Monte-Carlo simulation model of sequences of capital rationing investment decisions.

Deciding whether to manufacture a part in-house involves a make-or-buy decision analysis. If a number of parts are being considered, a decision on which parts to make and which to purchase is required. The complexity of the problem increases when the decision maker must decide which machines must be purchased and which part items will be assigned to each machine. Yenradee et al. [15] show that the capital budgeting analysis of machine selection and part assignment can be formulated as an integer linear programming problem in which the following set of decisions are to be made:

1. Which part items should be taken back for in-house manufacture?

2. What machine sizes and how many machines should be purchased?

3. And, what is the optimal part-machine assignment?

In their paper, they assume that the four maximum budget levels (i.e., 40 million baht, 45 million baht [baht is the currency of Thailand], 50 million baht, and unlimited budget, where 25 baht is equivalent to $1.00) they consider are fully available at the beginning of Year 1. This assumption may be invalid in a real situation, because obtaining a large loan can be difficult for several small- to medium-sized organizations.

We will revisit the same machine selection and part assignment problem by considering a more realistic budget constraint. Our objective is to develop a more practical capital budgeting policy that includes not only the previously mentioned three decisions, but also the timing of the capital investment. Three cases are considered. In Case I, we distribute the capital budgets over a 2-year period and consider that the machine investment costs and the purchase costs of parts increase every year. We also assume that a fixed percentage of the previous year's savings is reinvested in the following year in conjunction with that year's budget allocation. In Case II, we do not allow the use of a portion of the previous year's savings as an additional capital investment. Finally in Case III, we assume that a single large budget is available at the beginning of Year 1. For these three cases, the total amounts of the allocated budgets (without considering the time value of money) are equal.

PROBLEM DESCRIPTION AND COST DATA

The capital budgeting problem and its cost data we investigated in this paper are obtained from an international electric appliance manufacturing company in Thailand. The company manufactures household products, namely, television sets, electric fans, refrigerators, washing machines, and electric rice cookers. At present, most of the plastic parts are purchased from suppliers while some are produced at the company-owned plastic injection factory. Due to high cost, late delivery, and poor quality of purchased plastic parts, the company is considering an option to expand its Plastic Injection Division and produce more parts in-house. A preliminary screening (based on quantity, purchase cost, and production process) indicates that 20 part items currently being purchased from suppliers are possible candidates in a make-or-buy analysis. The analysis also shows that five injection machine sizes are needed for the in-house production of these parts. (Existing injection machines are operating at full capacity.) An injection machine can produce a variety of plastic part items depending on the design of the injection mold and the part size. Similarly, a plastic part item may be alternately produced by different injection machines. When using different machines to produce the same part item, the injection cost varies; that is, the injection cost of the bigger machine is more expensive then that of the smaller one. Hence, the company must decide which plastic part items to produce in-house, which machine size(s) and how many to purchase, and what is the optimal machine-part assignment? Also, the company estimates the most likely amount of the capital budget that can possibly be allocated is 15 million baht for the first year. For the following year, the capital budget is reduced to 10 million baht. It also is expected that 80% of the annual savings obtained from the first year will be allocated to the following year for purchasing additional injection machines. This constraint introduces another question: What will be the optimal capital budgeting policy for these 2 years? To be specific, what machine size(s) and how many should be purchased, when should they be purchased, and which part items should be assigned to each machine?

TABLE 1 shows the capital investment required for individual machines. The value is based on the total purchase cost (which includes cost, insurance, and freight [CIF], import duty, seaport fee, installation cost, and construction cost (for additional floor support and other supporting facilities).

The CIF is expected to increase annually. The increase rate, however, varies with the machine size. Here we assume that the increase rate for 350T and 220T is 15% per year while that for 850T, 650T, and 170T is 10% per year. The increase in CIF affects both the duty fee and installation cost. Similarly, we assume that the construction cost increases at 5% per year. TABLE 2 shows the total cost of five plastic injection machine sizes for Year 1 and Year 2.

[TABULAR DATA FOR TABLE 1 OMITTED]

TABLE 2. Estimated costs of injection machines for Year 1 and
Year 2.

                              Cost (baht)
Machine      Size
                          Year 1          Year 2

1            850T       12,231,000      13,374,000
2            650T        9,510,000      10,399,000
3            350T        5,216,000       5,930,000
4            220T        3,453,000       3,926,000
5            170T        2,906,000       3,179,000

CIF increases 10% per year for 850T, 650T, and 170T, and 15% per
year for 350T and 220T.

The construction cost increases 5% per year.

To determine the optimal part-machine assignment, it is necessary to know which injection machine(s) can be used to produce individual plastic part items. Additionally, the annual savings obtained and the total operating time for producing the required quantity are needed. The annual savings of the part is estimated from the difference between the annual purchase cost and the annual manufacturing cost (which includes raw materials cost, labor cost, injection cost, and other allocated operating expenses). For the same part item, the annual savings decrease with an increase in the size of the injection machine assigned to produce it. To cope with the rising cost of purchased parts, we assume that the annual savings for each part item increases at a rate of 5% per year. (The inflation rate and the income tax are not considered in this paper.)

The total operating time depends on the part item and its production quantity. TABLE 3 lists the annual savings and the required operating times of plastic parts for all injection machines being considered. For convenience, plastic part items are numbered from 1 to 20. Similarly, the injection machines are numbered from 1 to 5, in which No. 1 represents 850T and No. 5 represents 170T. The maximum available operating time for a machine is 24 days/month. Thus, part items in which their required operating times are greater than 24 days/month need multiple units of the same injection machine.

TABLE 3. Injection machine to be assigned, annual savings, and
operating time of plastic parts.

Part      Possible         Annual Savings(a)          Operating
Number    Machine                                      Time(b)

1              1               1,194,000                 1.87
2              1               2,060,000                 1.87
3             1, 2      (1,562,000), (1,620,000)    (2,63), (2.63)
4             1, 2      (1,552,000), (1,609,000)    (2.63), (2.63)
5             1, 2      (1,383,000), (1,904,000)    (23.84), (23.84)
6              3                730,000                   6.02
7              3                729,000                   6.02
8              3                842,000                   7.97
9              3                788,000                   8.57
10             3                792,000                  18.78
11             3                954,000                  38.15
12             4                590,000                   6.06
13             4              1,003,000                  12.27
14             4                979,000                  12.27
15            4, 5      (1,057,000), (1,095,000)     (6.99), (6.99)
16            4, 5      (1,367,000), (1,427,000)    (10.97), (10.97)
17            4, 5        (845,000), (883,000)        (6.99), (6.99)
18            4, 5        (875,000), (917,000)        (7.81), (7.81)
19            4, 5      (1,108,000), (1,170,000)     (11.23), (11.23)
20            4, 5        (922,000), (1,007,000)     (15.44), (15.44)

a All annual savings are in baht (25 baht is approximately $1.00).

b The unit of the operating time is days/month.

c For parts that can be manufactured by two machine sizes, the first
value belongs to the first machine, the other belongs to the second.

In the next section, we will develop the general mathematical model of the problem that consists of the objective function and the constraints. Then we will use this model to describe our case study examples and solve them using an optimization software tool called LINGO.

PROBLEM FORMULATION

To formulate the capital budgeting problem, we used the following notation:

m = Number of machine sizes being considered, units.

n = Number of plastic part items being considered, items.

I = Discount rate, %.

g = Annual increase of savings, %.

t = Useful life of plastic injection machines, years.

l = Length of investment period, years (1 [less than or equal to] t).

[S.sub.ijk] = Savings at end of Year k, obtained when assigning part i to machine j during Year k.

S(k) = Total savings in Year k (at end of the year).

[C.sub.jk] = Investment cost required for purchasing one unit of machine j during Year k (at beginning of year).

[N.sub.jk] = Nonnegative integer variable representing number of machine j purchased at beginning of Year k, units.

[M.sub.t] = Operating time required to produce required quantity of part i, days/month.

[X.sub.ijk] = Binary variable, {0,1}, being 1 if part i is produced by machine j during Year k; being 0 otherwise.

PART(j) = Set of plastic parts that machine j can produce.

PTMC = Set of part-machine assignment combinations.

MC(i) = Set of injection machines that can produce part i.

For simplicity, the following conditions are assumed.

1. The discount rate is constant throughout the entire study.

2. All machines have equal useful life.

3. The straight-line depreciation method is used. The final salvage value is assumed to be zero at the end of the useful life.

4. The study period is equal to the machine's useful life. At the end of the study period, those machines that have remaining value will be sold at their predicted book values.

5. The annual allocated investment budget can be of any amount. However, it is required that from Year 2 to the last year of the investment period, the amount of the budget will be constant throughout. (All machine purchases are made at the beginning of the year.)

6. A fixed percentage of the annual savings earned during the year will be used as an additional capital investment for the following year, if the machine investment is allowed for the next year.

7. During the investment period, the part-machine assignment is not fixed. The assignment will be permanent only when no more machines are purchased (i.e., when the investment period is over).

OBJECTIVE FUNCTION

The objective function has three cost components: total savings, total machine investment and total salvage value. Note that the economic criterion used in this study is the net present value (NPV).

NPV of the Total Savings

During the first I years, the annual savings varies with the size and the number of the injection machines purchased and the part-machine assignment. For the same part item, the savings increases at a rate of g per year. Generally, the savings in Year k, S(k), can be written as

[Mathematical Expression Omitted] (1)

From Year (l + 1) onward, no additional machines will be purchased. The total annual savings increases at a rate of g per year. We can depict the cash flow of this series of annual savings by the geometric series such that

[Mathematical Expression Omitted], (2)

where P = the present value of the total savings of the series, S(l) = the first annual savings in the series, and g[prime] = 1+I/1+g - 1. See Thuesen and Fabrycky [15] for the derivation.

Subsequently, we can determine the NPV of the total savings, NPV(S), from

[Mathematical Expression Omitted], (3)

where

[Mathematical Expression Omitted],

[Mathematical Expression Omitted],

g[prime] 1 + I/1 + g - 1, and k = 1, 2, ..., l

NPV of the Total Machine Investment

Since all machine purchases will occur only during the investment period, the NPV of the total machine investment, NPV(M), can be written as

[Mathematical Expression Omitted]. (4)

NPV of the Total Salvage Value

Since the machine's useful life coincides with the length of the study period, the machines purchased at the beginning of Year 1 will have zero salvage value at the end of Year t. Those purchased afterward (i.e., during Year 2 to Year 1), however, still have their book values at the end of the study period. Using the straight-line depreciation method, we can express the NPV of the total salvage value, NPV(SV), as

[Mathematical Expression Omitted]. (5)

Letting z be the NPV of the total "net" savings, we will have

z = NPV(S) +NPV(SV) - NPV(M). (6)

CONSTRAINTS

This capital budgeting problem has three constraints: mutually exclusive, available machine time, and allocated annual budget. The formulation of these constraints is discussed below.

Mutually Exclusive Constraint

It is assumed that there is only one injection mold for each part. The mutually exclusive constraint therefore states that when a part item has been assigned to a specific machine, it cannot be assigned to others. Mathematically, this constraint can be written as

[summation of] [X.sub.ijk] [less than or equal to] 1, where j [element of] MC(i) for i = 1, 2, ..., n; and k = 1,2, ..., l. (7)

Machine Time Constraint

All injection operations assigned to a machine can be performed only in the available time of the machine. Letting [Y.sub.j] be the available operating time of one unit of machine j, we then obtain

[Mathematical Expression Omitted], for j = 1,2, ..., m; and k = 1,2, ..., l. (8)

Note that the term [Mathematical Expression Omitted] represents the cumulative number of available machine

j's at the beginning of Year k.

Allocated Annual Budget Constraint

For simplicity, we will assume that the allocated budget for Year 1 is [V.sub.1] and those for Year 2 to Year I are fixed at V. Furthermore, a fixed percentage b of the annual savings is added to the allocated investment budget of the following year. Thus, the budget constraint can be expressed as

[Mathematical Expression Omitted], for k = 1, 2, ..., l. (9)

In summary, the capital budgeting problem with limited annual budget can be described as

maximize = [Mathematical Expression Omitted],

subject to

[summation of] [X.sub.ijk] where j [element of] MC(i) [less than or equal to] 1,for i = 1, 2, ..., n; and k = 1, 2, ..., 1,

[Mathematical Expression Omitted], for j = 1, 2, ..., m; and k = 1, 2, ..., l,

[Mathematical Expression Omitted], for k = 1, 2, ..., l,

[X.sub.ijk] + {0, 1}, [for every] ijk,

and

[N.sub.jk] = nonnegative integer, [for every] jk,

where

[Mathematical Expression Omitted],

[Mathematical Expression Omitted], and k = l,2, ..., l.

Case I: The budget is spread over 2 years with a portion of the previous year's savings used as additional capital investment.

In our machine selection and part assignment problem, the following conditions and data are used:

1. The discount rate is 20% and is constant throughout the entire study.

2. The useful life of all injection machines is 13 years.

3. The investment period is 2 years. The maximum capital investment for Year 1 is 15 million baht. For the following year, an additional 10 million baht investment budget is allocated.

4. For each part item, the savings due to an in-house production increases at 5% per year.

5. The required capital investment of each injection machine for Year I and Year 2 follows what is listed in TABLE 2.

6. Eighty percent of the total savings earned during the first year will be used as additional allocated capital investment for the following year.

7. The available operating time of one unit of the injection machine is 24 days/month, irrespective of machine size.

The optimal machine selection and part assignment can then be obtained by solving the following Case I problem.

[Mathematical Expression Omitted],

subject to

[Mathematical Expression Omitted], for i = 1, 2, ..., 20; and k = 1 and 2,

[Mathematical Expression Omitted], for j = 1, 2, ..., 5; and k = 1 and 2,

[Mathematical Expression Omitted], for k = 1 and 2,

[X.sub.ijk] = {0,1} [for every] ijk,

and

[N.sub.jk] = nonnegative integer, [for every] jk,

where

[Mathematical Expression Omitted], S(0) = 0, and k = 1 and 2.

Case II: The budget is distributed over 2 years without a portion of the previous year's savings used as additional capital investment.

The Case II problem is similar to that described in Case I except that no percentage of the first year's savings is to be used as an additional machine investment in the second year. The problem formulation in this case is essentially the same as that in the first case. The only difference is that the last term in the allocated annual budget constraint disappears.

Case III: The single large budget is available at the beginning of the project.

In Case III, the following changes in conditions and data are assumed:

1. The investment period is 1 year. The maximum allocated single budget is 25 million baht and is available at the beginning of the project (i.e., at the beginning of Year 1).

2. All machine purchases are allowed only at the beginning of Year 1.

3. The part-machine assignment once determined (at the beginning of Year 1) remains unchanged throughout the entire study period.

4. The NPV of the total salvage value is zero because all machines are depreciated to zero at the end of the study period.

Other conditions and data such as the discount rate, the escalation rate of annual savings, and the machine's useful life are unchanged.

From the above conditions and data, the reduced model for Case II is

maximize = [Mathematical Expression Omitted]

subject to

[summation over j [element of] M C(i)] [X.sub.ij] [less than or equal to] 1, for i = 1,2, ..., 20,

[summation over i [element of] PART(j)] [M.sub.i][X.sub.ij][less than or equal to] 24[N.sub.j],for j = 1,2, ..., 5,

[summation of] [N.sub.j][C.sub.j] where j = 1 to 5 [less than or equal to] 25 x [10.sup.6]

[X.sub.ij] = {0,1}, [for every]ij,

and

[N.sub.j] = nonnegative integer, [for every j].

RESULTS AND DISCUSSION

Solving the machine selection and part assignment problem described in Case I with an integer linear programming technique yields the maximum total net savings of 61,659,000 baht. The results are summarized in TABLES 4 and 5. Among the five injection machine sizes being considered, only four sizes are recommended for purchase. They are 850T, 350T, 220T, and 170T. for the machine size with multiple units, the timing of purchase of each unit is different. For example, the first unit of the 220T Machine is to be purchased at the beginning of the first year while the second unit will be purchased at the beginning of the second year. At the end of Year 2, the total number of machines purchased is 6 units: 1 unit of 850T, 1 unit of 350T, 2 units of 220T, and 2 units of 170T. The NPV of the total machine investment is 28,898,000 baht. TABLE 4 shows the recommended years for purchasing them.

TABLE 4. Machine size, number of units purchased, and investment
cost per year (Case I).

          Machine      Number Total
Year
          Size           of Units      Investment(*)

1         350T              1           14,481,000
          220T              1
          170T              2

2         850T              1           17,300,000
          220T              1

NPV of Total Investment (baht)          28,898,000
NPV of Total Net Savings (baht)         61,659,000

* The total investment (baht) is the actual cost at the beginning of
the year.

TABLE 5 shows the assignment of plastic parts to different injection machines in each year. The number of parts selected for in-house production increases when the allocated budget increases. Their cumulative numbers are 10 and 16 part items in Year 1 and Year 2, respectively.

TABLE 5. Optimal part-machine assignment(a) (Case I).

        Machine
Year                               Assigned Parts
        Size       Units      (and Machine Utilization)

1         350T       1        6, 8, 9                     (94.00%)
          220T       1        12, 13                      (76.38%)
          170T       2        15, 16, 17, 18, 19          (91.65%)

2         850T       1        1, 2, 3, 4                  (37.50%)
          350T       1        6, 8, 9                     (94.00%)
          220T       2        12, 13, 14, 15, 17          (92.88%)
          170T       2        16, 18, 19, 20              (94.69%)

a After Year 2, the part-machine assignment remains unchanged
because no additional machines will be purchased.

Several interesting findings deserve detailed discussions. First, a part-machine reassignment occurs to many part items. In this situation, a specific part is assigned to a certain machine and later reassigned to another machine in the next year. See TABLE 3 for a list of machines that can be assigned to produce the part. The purpose is to maximize the total net savings earned during the year. (It is based on the assumption that no additional cost is incurred when switching the machine for the plastic part injection.) Parts No. 15 and No. 17 are examples of this part-machine reassignment. During Year 1, these parts are to be produced by the 170T Machine. Then in Year 2, when another 220T Machine is recommended for purchase, they are reassigned to the new 220T Machine. This part-machine reassignment increases the average utilization rate of the 220T Machine (from 76.38% to 92.88%).

Second, several new parts are recommended to be taken back at the beginning of Year 2 either as a result of a new machine purchase (i.e., the 850T Machine) or a part-machine reassignment. When the 850T Machine is purchased, the Plastic Injection Division can take back four additional parts: Parts No. 1, 2, 3, and 4. This decision is made because the amount of savings obtained from producing these four parts is significantly large, even though the utilization rate of the 850T Machine is low (37.50%). Similarly, we can see that when the new 220T Machine is to be purchased at the beginning of Year 2, Parts No. 15 and 17 are reassigned to that machine size, allowing a new part, Part No. 20, to be taken back. Assigning Part No. 20 to the 170T Machine also increases the average utilization rate from 91.65 % to 94.69%.

Note that in our analysis, we assume that the unused budget will be reallocated to the Plastic Injection Division in the following year. Although this unused fund generates the interest at a rate of 20% per year, the earned interest will not be allocated to the Division. Thus, we ignore its computation in EQUATION 9.

In Case II, we remove the contribution of the first year's annual savings from the allocated annual budget. The results show that four injection machines are recommended for purchase. They consist of 1 unit of the 850T Machine, 1 unit of the 350T Machine, and 2 units of the 170T Machine. The NPV of the total net savings is 50,709,000 baht. Additionally, the NPV of the total machine investment is 22,173,000 baht. TABLES 6 and 7 summarize the machine purchases and the part-machine assignments in both years. The cumulative numbers of parts taken back in Year 1 and Year 2 are 8 and 12 units, respectively.

TABLE 6. Machine size, number of units purchased, and investment
cost per year (Case II).

           Machine       Number          Total
Year
            Size        of Units      Investment(*)

1           350T           1            11,028,000
            170T           2

2           850T           1            13,374,000

NPV of Total Investment (baht)          22,173,000
NPV of Total Net Savings (baht)         50,709,000

* The total investment (baht) is the actual cost at the beginning
of the year.
TABLE 7. Optimal part-machine assignment(a) (Case II)

        Machine
Year                               Assigned Parts
         Size      Units      (and Machine Utilization)

1        350T        1        6, 8, 9                      (94.00%)
         170T        2        15, 16, 17, 18, 19           (91.65%)

2        850T        1        1, 2, 3, 4                   (37.50%)
         350T        1        6, 8, 9                      (94.00%)
         170T        2        15, 16, 17, 18, 19           (91.65%)

a After Year 2, the part-machine assignment remains unchanged
because no additional machines will be purchased.

When comparing the results shown in TABLES 5 and 7, especially in Year 2, we see that for the same machine size, the average machine utilization rates are the same for the 850T Machine and 350T Machine because their part-machine assignments are identical. For the 170T Machine, its utilization rate in Case I is slightly higher than that in Case II. Moreover, the first year's and the second year's machine investments in Case II are to be only 73.52% and 95.72%, respectively, of the available budget as compared to 94.54% and 98.54% observed in Case I. Adding these findings to the greater NPV of the total net savings seen in Case I, we see clearly that the investment policy used in Case I is undoubtedly superior to the one used in Case II.

In Case III, we assume that a single large budget is available at the beginning of Year 1. We also assume that a series of machine purchases in different years no longer exists. All machine purchases and the part-machine assignment must be finalized at the beginning of Year 1. This assignment will then remain unchanged for the entire study period.

By simplifying the general model of the capital budgeting problem, we see that the simplified (reduced) model reflects the following changes.

1. The term in the objective function (z) that represents the NPV of the total salvage value disappears because all machines are purchased at the beginning of Year 1.

2. The allocated budget of the first year ([V.sub.1]) is 25 million baht.

3. An additional 10 million baht investment budget and a contribution of the previous year's annual savings also disappear from the allocated budget constraint.

4. The subscript k is removed from the model because all machine purchases must be made at the beginning of Year 1; that is, k = 1.

Solving the reduced model by LINGO yields the NPV of the total net savings of 59,420,000 baht. The analysis shows that five machines are recommended: 850T (1 unit), 220T (2 units), and 170T (2 units). The total machine investment is 24,949,000 baht. TABLES 8 and 9 show the results of Case III.

TABLE 8. Machine size, number of units purchased, and investment
cost (Case III)

          Machine       Number          Total
Year
           Size        of Units      Investment(*)

1          850T            1           24,949,000
           220T            2
           170T            2

NPV of Total Investment (baht)         24,949,000
NPV of Total Net Savings (baht)        59,420,000

* The total investment (baht) is the actual cost at the beginning of
the year.
TABLE 9. Optimal part-machine assignment(a) (Case III)

          Machine                       Assigned Parts
Year
           Size       Units       (and Machine Utilization)

1          850T         1         1, 2, 3, 4              (37.50%)
           220T         2         12, 13, 14, 15, 17      (92.88%)
           170T         2         16, 18, 19, 20          (94.69%)

a After Year 1, the part-machine assignment remains unchanged
because no additional machines will be purchased.

Comparing the results shown in TABLES 5 and 9 (for Case I and Case HI, respectively), we observe the following. First, the machine utilization of the same machine type (size) is the same for both cases. This consistency occurs because the same groups of part items are assigned to individual machine sizes. This finding could indicate that such part-machine assignment is the optimal assignment. Therefore, when a new machine is recommended (in Case I), a new group of part items is assigned to it.

Second, we see that the optimal solution obtained in Case I (61,659,000) is greater than that in Case III (59,420,000), which indicates that spreading the budget over years is superior to investing the single amount in 1 year. Initially, we have anticipated the opposite outcome because with the annual increases in savings and costs of machines - if the fund is sufficient - investing as much as possible is logical as long as the NPV of the cash flow is positive. We attribute this interesting finding to our assumption that 80% of the savings obtained in Year l can be allocated as an additional investment budget for Year 2. Without this extra budget (as seen in Case II), we find that Case III becomes superior because when adding the 10 million baht (after being discounted by 20%) to the initial 15 million baht, the total allocated budget will be less than 25 million baht that is assumed in Case III. Note that the remaining cash inflow (20% of the savings in Case I and 100% in Case II) will earn a lower rate of return (minimum attractive rate of return [MARR] = 20%) than when they are allocated as an additional investment budget.

Further detailed analyses show that each year, the company can invest up to almost 100% of its allocated budget. In Case I, the first-year investment is 14,481,000 baht (96.54%) while the second-year investment is 17,300,000 baht (98.54%). Note that the second-year budget is the sum of 10 million baht and 80% of the first-year savings (which is 9,445,000 baht). In Case III, the total machine investment is 24,949,000 baht (99.80%).

CONCLUSIONS

In this paper, we have developed a general mathematical model that can assist management to decide between the "make" and the "buy" options and to determine the optimal part-machine assignment. The model can be used in situations such as when a single large budget is available, when the budget is distributed over a number of years, and when a percentage of savings obtained from the previous year can be used as an additional investment budget. It also accounts for realistic variables such as the length of the investment period, the escalation rate of the annual savings and the costs of machines, the required machine operating time, and the effect of the remaining salvage values. The model provides the following solutions: the types of machine and the number of units to be purchased, the timings of purchase, and the part-machine assignment for each year. Decision makers can vary the variables to determine the optimal recommendations for each situation. It thus is as an effective tool for solving complex machine selection and part assignment problems. At present, some reasonably prices optimization software packages are available. They can solve this type of capital budgeting problem, if the number of parameters does not exceed the software package's limitation and/or the computer's capability.

Based on the case study examples and assumptions, we found that distributing the machine investment over a number of years and adding a portion of the previous year's savings to the allocated budget are superior to investing the equal sum in 1 year. However, we do not recommend generalizing these findings to cover every situation. Variables that may significantly influence the results are the length of the investment period, the amount of the annual investment budget, and the percent contribution of the previous year's savings to the current year's budget. A sensitivity analysis can be done to determine which variable(s) play a dominant role in capital budgeting decision making.

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BIOGRAPHICAL SKETCH

SUEBSAK NANTHAVANIJ is an Associate Professor of Industrial Engineering at Sirindhorn International Institute of Technology, Thammasat University, Thailand. He received his B.S. in chemical engineering from Chulalongkorn University, Thailand, and both M.S. and Ph.D. in industrial engineering from the University of Texas, Arlington, Texas. He received a 1996 Fulbright scholarship to serve as a senior visiting scholar at New Jersey Institute of Technology, Newark, N.J. Dr. Nanthavanij is an associate editor of Engineering Design and Automation, Journal of Engineering Valuation and Cost Analysis, International Journal of Industrial Engineering, and Journal of Industrial Engineering Research. He is a senior member of IIE and a member of the Human Factors and Ergonomics Society, the IEEE Computer Society, the South East Asian Ergonomics Society, and the Engineering Institute of Thailand.

PISAL YENRADEE is an Associate Professor of Industrial Engineering at Sirindhorn International Institute of Technology, Thammasat University, Thailand. He received his B.Eng. in production engineering from King Mongkut's Institute of Technology (North Bangkok), Thailand, and both M.Eng. and D.Eng. in industrial engineering from the Asian Institute of Technology, Thailand. Dr. Yenradee is a member of IIE and the Engineering Institute of Thailand. His research interests include production management, inventory control, and systems simulation.